Short-term prediction of the behavior of a dynamic system using fuzzy Markov chains

Abstract

The problem of predicting the behavior of a dynamic system with sharply changing stochastic properties is considered. The dynamic system is represented by the corresponding numerical time series of system indicators. Solving such a problem requires determining the moments in time of change in stochastic properties, disordering the behavior of the time series, and identifying time segments with homogeneous behavior. Statistical models for short-term forecasting can be built on such segments. For short-term forecasting, it is proposed to use a discrete Markov chain model for fuzzy states instead of numerical statistical models such as ARIMA and recurrent neural networks. In this case, the regression problem is replaced by a fuzzy classification problem and the object of study becomes the time series of fuzzy states generated by the original numerical time series. This approach simplifies obtaining a solution and increases the reliability of the forecast. An algorithm for recurrent estimation of the stochastic matrix of a Markov chain model with fuzzy states has been developed. To identify homogeneous segments of a time series of fuzzy states, it is proposed to calculate a vector of stationary probabilities (eigenvector of the matrix) at each step of the recurrent estimation of a stochastic matrix and, based on an analysis of its behavior, to identify homogeneous segments. The efficiency and effectiveness of the proposed approach is illustrated by examples of short-term forecasting of the behavior of time series of rates of various assets of the Moscow Exchange.